One-parameter isospectral special functions

"Using a combination of the ladder operators of Pina [1] and the Pammetric operators of Mielnik [2] we introduce second order linear differential equations whose eigenfunctions are isospectral to the special functions of the mathematical physics and illustrate the method with several key examples.""Usando una combinación de los operadores de escalera de Pina [1] y de los operadores parametricos de Mielnik [2] introducimos operadores lineales de segundo orden con eigenfunciones que son formas isoespectrales de las funciones especiales de la física matemática y presentamos algunos ejemplos básicos.

Eigenvalue problems, spectral parameter power series, and modern applications

"Our review is dedicated to a wide class of spectral and transmission problems arising in di?erent branches of applied physics. One of the main di?culties in studying and solving eigenvalue problems for operators with variable coe?cients consists in obtaining a corresponding dispersion relation or characteristic equa-tion of the problem in a su?ciently explicit form. Solutions of the dispersion relation are the eigenvalues of the problem. When the dispersion relation is known the eigenvalues are found numerically even for relatively simple problems with constant coe?cients because even in those cases as a rule the dispersion relation represents a transcendental equation the exact solutions of which are unknown.

An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators

"We introduce an alternative factorization of the Hamiltonian of the quantum harmonic oscillator which leads to a two-parameter self-adjoint operator from which the standard harmonic oscillator, the one-parameter oscillators introduced by Mielnik, and the Hermite operator are obtained in certain limits of the parameters. In addition, a single Bernoulli-type parameter factorization which is different of the one introduced by M. A. Reyes, H. C. Rosu, and M. R. Gutie´rrez, Phys. Lett. A 375 (2011) 2145 is briefly discussed in the final part of this work.

Periodic Sturm-Liouville problems related to two Riccati equations of constant coefficients

"We consider two closely related Riccati equations of constant parameters whose particular solutions are used to construct the corresponding class of supersymmetrically-coupled second-order differential equations. We solve an-alytically these parametric periodic problems along the positive real axis. Next, the analytically solved model is used as a case study for a powerful numerical approach that is employed here for thefirst time in the investigation of the en-ergy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and imple-ments the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill dis-criminant based on on Kravchenko´s series.

Solutions of the perturbed KdV equation for convecting fluids by factorizations

"In this paper, we obtain some new explicit travelling wave solutions of the perturbed KdV equation through recent factorization techniques that can be performed when the coefficients of the equation fulfill a certain condition. The solutions are obtained by using a two-step factorization procedure through which the perturbed KdV equation is reduced to a nonlinear second order differential equation, and to some Bernoulli and Abel type differential equations whose solutions are expressed in terms of the exponential and Weierstrass functions.

Additive and multiplicative noises acting simultaneously on Ermakov-Ray-Reid systems

"We investigate numerically the effect of additive and multiplicative noises on parametric oscillator systems of Ermakov-Ray-Reid type when both noises act simultaneously. We find that the main perturbation effects on the dynamical invariant of these systems are produced by the additive noise. Different from the separate action of the multiplicative noise when the dynamical invariant of these systems is robust, we also find a weak effect that can be attributed to the multiplicative noise.""Se investigan numéricamente los efectos de los ruidos aditivos y multiplicativos sobre los sistemas dinámicos de osciladores paramétricos de tipo Ermakov-Ray-Reid cuando los dos tipos de ruidos actúan de manera simultánea. La mayor parte de la perturbación del invariante proviene del ruido aditivo. A diferencia del caso cuando el ruido multiplicativo actúa por separado y el invariante dinámico presenta robustez, encontramos que en la acción simultánea de los dos ruidos hay también un efecto pequeño atribuible al ruido multiplicativo.

Newton´s laws of motion in the form of a Riccati equation

"We discuss two applications of a Riccati equation to Newton´s laws of motion. The first one is the motion of a particle under the influence of a power law central potential V(r)=krε. For zero total energy we show that the equation of motion can be cast in the Riccati form. We briefly show here an analogy to barotropic Friedmann-Robertson-Lemaitre cosmology where the expansion of the universe can be also shown to obey a Riccati equation. A second application in classical mechanics, where again the Riccati equation appears naturally, are problems involving quadratic friction. We use methods reminiscent to nonrelativistic supersymmetry to generalize and solve such problems.

Adsorption of molecular gases on different porous surfaces using the statistical associating fluid theory variable range approximation

"In the thermodynamic framework of coupled statistical associating fluid theory variable range (SAFTVR) 2D and 3D models we present the theoretical predictions of the adsorption isotherms in real physical units, as it is commonly done in experiments. The systems studied are methane (CH4), nitrogen (N2), ethane (C2H6), n-butane (C4H10), propane (C3H8), and propylene (C3H6) adsorbed on silica gel (two classes: NSG and WSG), zeolite (4A and Na-Y), and BDH activated carbon. Employing only two fitting parameters with clear physical meaning in such an approach, we find a better agreement with the experimental data than other semiempirical models with more fitting parameters.

PI-controlled bioreactor as a generalized Liénard system

"It is shown that periodic orbits can emerge in Cholette’s bioreactor model working under the influence of a PI-controller. We find a diffeomorphic coordinate trans-formation that turns this controlled enzymatic reaction system into a general-ized Lie´nard form. Furthermore, we give sufficient conditions for the existence and uniqueness of limit cycles in the new coordinates. We also perform numerical simu-lations illustrating the possibility of the existence of a local center (period annulus). A result with possible practical applications is that the oscillation frequency is a function of the integral control gain parameter.

A simple electronic circuit realization of the tent map

"We present a very simple electronic implementation of the tent map, one of the best-known discrete dynamical systems. This is achieved by using integrated circuits and passive elements only. The experimental behavior of the tent map electronic circuit is compared with its numerical simulation counterpart. We find that the electronic circuit presents fixed points, periodicity, period doubling, chaos and intermittency that match with high accuracy the corresponding theoretical values.